# Memorizing Trig Identities

I adjunct for a local community college teaching College Algebra and College Trigonometry. Every year, the community college math department insists on students memorizing each of the trig identities. I can understand why students should have the pythagorean identities memorized and the unit circle memorized. But, why should a student have to memorize double angle, half angle, product to sum formulas, etc? I'm trying to get a better grasp on why the college requires this and will not allow students to have a notecard for these. I don't remember the product to sum formula being a pre-requisite for Calculus or any higher level courses. Does anyone have any insight on this, or is it just an odd request from the college?

• It seems odd to me that your college is trying to micromanage you to this extent. Academic freedom is not just for full-time faculty. At the community college where I teach, I have never heard of anyone being told that they have to have specific policies such as not allowing notecards on tests. In general, the reason many community college educators are in love with forced memorization is that memorization is easy, and understanding is hard. Enforcing memorization is a good way to create a Potemkin village impression that our students are learning and succeeding.
– user507
May 5, 2017 at 4:04
• I must differ with @BenCrowell's characterization. It is common for departments to agree on uniform final exams with uniform procedures; in places with many adjuncts this is a somewhat necessary check on quality. Memorization is often on our minds because most community college students do not have automaticity with grammar-school skills like times tables, negative numbers, order of operations, operations on fractions, factoring, etc.; which creates a barrier to following any higher-level derivations. May 5, 2017 at 5:45
• That said, I agree that memorizing trigonometric formulas is likely an unthinking legacy issue. When I teach trigonometry I certainly do provide a formula card. May 5, 2017 at 5:46
• When I took the NY Trig Regents (standardized NY high school exams), we had to memorize two pages of trig formulas. I memorized them before the test but I don't remember any of them. I can derive many of them though. May 5, 2017 at 7:18
• An old salt in my department had this to say on the matter, when he coordinated our precalc classes: If students are forced to memorize the identities, they'll at least remember that they exist by the time they need to use them in a calc class. So we require memorizing all but the sum-product conversion identities. May 13, 2017 at 11:05

I agree this memorization is not necessary.

If students understand how the trigonometric functions are defined (unit circle) and know several basic identities, everything else can be derived. I think the pythagorean identity and the sine and cosine of sum of angles are sufficient.

$$\sin^2 \alpha + \cos^2 \alpha = 1$$

$$\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)$$

$$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$$

Math is not about memorization. If students are taught the basics and learn how to derive whatever they need, everybody will be happier.

• In my personal education I found repeated derivation as needed to be the best way to memorize. May 4, 2017 at 23:56
• Math is not entirely not about memorization. People do need automaticity with some skills like times tables, negative numbers, fractions, decimals, percent, and the order of operations, or else they will not have the mental bandwidth to accomplish higher-order tasks. C.f., Alfred North Whitehead: "Civilisation advances by extending the number of important operations which we can perform without thinking about them." (madmath.com/2011/02/thoughts-and-cavalry.html) May 5, 2017 at 5:50
• I know I forgot things the first few times but remembered cos (a+b) was some combination of cos a, cos b, sin a and sin b. Then I worked it out with a=0, b=0, or a != 0/b=a, other values...this isn't rigorous derivation, but it helps someone who might be having trouble. It also may help to allow students a visual derivation (you can google for this). So it's nice to show/know several potentially simple ways to do this sort of thing. May 15, 2017 at 22:56
• The way I remembered and derived the functions-of-sum formulae was to memorise the one for $\sin(\alpha+\beta)$ (the symmetry helping some), then differentiate w.r.t. one of $\alpha,\beta$! (And formulae for differences are redundant since the above are valid for all values of the angles, positive and negative.) My TA hated when I did this, or maybe there was some misunderstanding/miscommunication involved. Sep 10, 2017 at 18:30

If you want to integrate $\sin^m x$ or $\cos^m x$ for even $m$, you need to reduce the powers, which requires some trigonometric identities beyond pythagorean identities (in this case, $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$). If the students are struggling with the application of trigonometric identities at this stage, then they are in trouble when they are asked to perform a "simplification" like the following before the calculus even begins:

$\sin^4 \theta = \left(\sin^2\theta\right)^2 = \left(\frac{1-\cos2\theta}{2}\right)^2 = \frac{1 - 2\cos2\theta + \cos^22\theta}{4} = \frac{1 - 2\cos2\theta + \frac{1+\cos4\theta}2}{4}$

A very very large number of students in my calculus classes do not get the "$4\theta$" at the end there, and the ones who are having trouble applying the identities in this way certainly never get the factor of $\frac14$ that comes from integrating such a thing, because they are buried in arcane symbols.

So I can imagine, at least for "double angle" or "half angle" formulas, that this line of reasoning would justify students committing these things to memory. If our calculus students see this "power-reducing" identity as "easy stuff I learned a long time ago," then they can much more easily reach the goals I've set out above.

That said, I don't know where to draw the line on memorization. But I can easily support drawing the line a little further out than just pythagorean identities and the unit circle.

• This just leads to the question: why do we think calculus students should be able to do this kind of calculation? I agree with you that this is (pretty much the only?) place that these formulas are needed. May 5, 2017 at 0:47
• And don't get me wrong: I love these formulas, because they encode some pretty cool math. In particular, you can prove the angle sum formulas only by using linearity of rotation operators. That is a neat story. You can use double angle formulas and tangent line approximations to compute sine and cosine by hand. That is a cool story too. I just do not really get to excited about being about to integrate $\sin^4(\theta)$ May 5, 2017 at 0:49
• The fact that the plots of $\sin^2\theta$ and $\cos^2\theta$ are identical to plots of $(1/2)\sin 2\theta$ and $(1/2)\cos 2\theta$, plus a constant $1/2$, is important for understanding applications like AC electrical power and energy in mechanical vibrations. One could debate whether students need to memorize the precise formulas, but IMO they certainly ought to understand the basic idea - and if they do understand it, they can easily obtain the precise formulas by drawing a sketch plot. May 5, 2017 at 1:34
• The sad state of affairs of calculus teaching these days, is that students who pass a calculus class might be able to find the above expression for the antiderivative of $\sin^4t$, while not having a clue about what an integral is. May 5, 2017 at 13:10
• It has to be reintroduced, but the familiarity helps. Students don't do well with two radically new things at the same time. May 16, 2017 at 16:41

If you have lots of time then you can surely derive most of the identities from the basics but basically it slows down your learning when these identites are required. Also not memorising these will give you a huge disadvantage in exams.

Another reason: sometimes it is required to know the form of the result of applying an identity quickly. Like while doing integration probably you will want to know that $\sin(x)\cos(y)=\text{some constant}*\text{sum of trig functions}$. You cannot just keep deriving identities and checking weather they will work or not. In other words you must know which identity to use and when and how.

Lastly I want to tell you my approach for memorising these:

(i) Everyday take one identity to be memorised
(ii) Derive it once in the beginning
(iii) Solve many questions related to it
(iv) If you forget it in between, rederive it
(v) repeat from step (iii) until it is memorised.

The advantage of this approach is that it will give you a solid understanding of the identity and also you will know where and when to apply it.

In fact I had never lerned the identities in the trig class .. i somehow got away with it. But when i went further and studies coordinate geometry, conic sections, integration, inverse trigonometry, etc. then I realised the importance of memorising them.

• Easy to integrate by substitution though ...But I do agree memorizing helps a lot.. Sep 15, 2017 at 21:17

Knowledge is power. The more trigonometry you can derive without needing to consult Google etc the better you can do on tests and, more importantly, the faster you can follow later conversations in calculus and differential equations and so forth. Here I assume the student embraces the idea that mathematical knowledge does not exist in isolation. Every new idea should be contrasted and compared and fit into the tapestry of their existing knowledge. In other words, I encourage the radical idea that college students should be scholarly. I know, this is at odds with the ever encroaching idea that we should make college as easy as possible for customers (students). Of course, if we allow students to look up every basic fact of trigonometry then it does make the course friendlier, but, when the course is about trigonometry it is (in my opinion) natural to expect the students to learn all the basic trig. identities.

Let me discuss here a method to derive such identities, brevity being the soul of wit, draw your own conclusions.

I would like to see us teach how $e^{i\theta} = \cos \theta + i \sin \theta$ naturally encodes just about every trig. identity you run into as a mere consequence of algebra. For example, $$e^{ia}e^{ib} = e^{i(a+b)}$$ is tantamount to the adding angles formulas for sine and cosine. It's not hard to derive $\cos x = \frac{1}{2}(e^{ix}+e^{-ix})$ and $\sin x = \frac{1}{2i}(e^{ix}-e^{-ix})$ and use them to calculate things like: \begin{align} \sin(a)\cos(b) &= \frac{1}{2i}(e^{ia}-e^{-ia})\frac{1}{2}(e^{ib}+e^{-ib}) \\ &= \frac{1}{2i}\frac{1}{2}(e^{i(a+b)}-e^{-i(a+b)}+e^{i(a-b)}-e^{-i(a-b)}) \\ &= \frac{1}{2}\underbrace{\frac{1}{2i}(e^{i(a+b)}-e^{-i(a+b)})}_{\sin(a+b)} + \frac{1}{2}\underbrace{\frac{1}{2i}(e^{i(a-b)}-e^{-i(a-b)})}_{\sin(a-b)} \end{align} Therefore, $\sin(a)\cos(b) = \frac{1}{2}\sin(a+b)+ \frac{1}{2}\sin(a-b)$. Of course, this is just a token example, there is so much more you can do if you learn this way of thinking.

My Modern Physics professor (Stephen Reynolds) mentioned that he would like to show us imaginary exponentials, but, he was either forbidden or discouraged (I forget, it's been a few years) from doing it, so, instead he'd just use real trigonometry. In retrospect, I really wish he had ignored whoever had given that advice. Imaginary exponentials make trigonometry into algebra.

I know that insightful application of the adding angles identities for sine and cosine can also derive very many things, but, it seems to me that technique is far more clever than the one I outline here. To use the adding angles formulas you have to come towards your goal in a sort-of sideways fashion. For example, $$\cos(x+x) = \cos x \cos x - \sin x \sin x = \cos^2 x-(1- \cos^2 x)$$ gives us a path to derive $\cos^2 x = \frac{1}{2}(1- \cos (2x))$. In contrast, to derive this with imaginary exponentials I just begin with my target $\cos^2 (x)$, do algebra, and find $\cos^2 x = \frac{1}{2}(1- \cos (2x))$.

So, why are we afraid of imaginary exponentials? I think the fear is real, but, is it rational? I wish we would all take Gauss' advice and get over this terminology "imaginary". Accept $\mathbb{C}$ as an integral and important part of basic school mathematics.

• I guess a counterargument would be that memorizing might prevent possible errors (provided of course that one memorized the correct formulas). Although I think that memorizing formulas might be helpful, it probably shouldn't be used as a crutch.
– K.M
Aug 2, 2019 at 21:58
• "So, why are we afraid of imaginary exponentials? I think the fear is real"... ugh... have a +1. anyways. Jan 5, 2020 at 22:41

If you see it filtering out students differentially by race or gender, you could discuss it in your department and college as an equity issue. I have taught trig dozens of times, and still do not have those memorized. (I can derive them easily.)

Math courses being used to filter students out is quite problematic, and equity concerns are one way we are making math courses fairer.

Students cannot be forced to memorize trig identities unless they are tested orally. Some of the students will and should learn to quickly derive some of them.
The ability to derive identities quickly as needed is a valuable skill. But tell them honestly that they can answer the exam questions correctly by either memorization or derivation of identities.

Students who study calculus will benefit from recognizing parts of identities. This is especially evident in learning integration techniques.

I teach mathematics. I am not in any of the other professions that use trigonometry, so I don't know which particular identities are needed in those fields But I do know that many use trigonometry in their jobs. I assume that they have need for some identities.

• I basically agree, but the type of fluency required to almost instantly derive the identities on an exam is a form of memorization. If you asked me what the 4th line of Robert Frost's poem "Stopping by the Woods on a Snowy Evening" was, I honestly wouldn't know -- until I traced through the poem mentally, at which point I would know ("To watch his woods fill up with snow"). Similarly, fluent identity derivation is more like tracing through a familiar story than it is puzzling through things from first principles. Sep 9, 2017 at 12:02

I think that it's an odd request from the college, for several reasons:

• I cannot remember any situation where I've needed these formulas, except perhaps for one or two integration exercises. (And then it's again the question: How many people are still solving their integrals manually in practice?)

• If I need them, I can find them in any math formula collection.

• If I need them and if I don't have a math formula collection available, I can easily derive them from the identities $e^{i\theta} = \cos\theta + i \sin\theta$ and $\sin^2 \theta + \cos^2 \theta = 1$. (These are the two formulas that one should memorize!)

The formulas may be more important for some people working in physics or electrical engineering, but these people need complex numbers anyhow.

I really believe that there are more important things that students should learn in mathematics, say, mathematical proof techniques.

Disclaimer:

Nowadays, I must be honest that I usually don't memorize any of the trig identities, save for possibly the Pythagorean identity because it appears so often. Instead I focus on remembering:

$$e^{i\theta}=\cos(\theta)+i\sin(\theta)\\\cos(\theta)=\Re(e^{\pm i\theta})=\frac{e^{i\theta}+e^{-i\theta}}2\\\sin(\theta)=\pm\Im(e^{\pm i\theta})=\frac{e^{i\theta}-e^{-i\theta}}{2i}$$

From these three identities alone, one can derive not only all of the standard identities, but many other niche identities that are not obvious at first glance. Common examples include:

• The general formulas for $$\cos^n(\theta)$$ and $$\sin^n(\theta)$$ in terms of multiple angles via the binomial theorem.

• The general formulas for $$\cos(n\theta)$$ and $$\sin(n\theta)$$ as factoring problems aside from the usual "repeatedly applying sums of angles".

• Sums of trig functions over angles in arithmetic progression via geometric series.

Furthermore, the procedure is straightforward algebra. Consider the example identity, supposing we don't get division by zero:

$$\frac{\cos[(n+1)\theta]-\cos(n\theta)}{\cos(\theta)-1}=\frac{\sin[(n+\frac12)\theta]}{\sin(\frac12\theta)}$$

I'd wager that trying to prove this identity with trig identities is a very troublesome task. However, with the complex exponentials, this problem is as easy as simply multiplying both sides out. You don't have to pick out which identities to use or how to use them because everything flows through the algebra you've built up.

1. I think there may be some differences in the right approach for a community college student versus for a math grad student. We are talking about very different intrinsic skill levels. Maybe the smarter students can derive more and the du...less smart students can derive less.

2. Not sure exactly how memorization is enforced. Verbal recital? If it is on problems, what is to stop from just fast deriving it (for the kids that can).

3. Myself, I found some identities not worth remembering (any of the squared ones, can pretty much be derived from sinsq plus cossq. But I did find a few identities, that it just seemed better to memorize (maybe the sum of sins and cosine angles). From that, you can get to double angles or differences of angles (just a different case of what you sum). And I had seen the deriviation of the initial identity. But the deriviation didn't stick as much, so I just committed the sum angles to memory.

4. I actually don't think it is the end of the world to memorize a page of trig identities. It comes pretty natural if you do it in concert with heavy drill to use the identities. And the student can kind of figure out per (3), which ones to memorize and which to re-derive during exams. I guess, basically I am more to the derive bias than the memorize, but it is not an all or nothing. And the key in either case, is heavy drill.

• -1; the wording of your point 1. is not constructive at all, and classifies/stereotypes student populations in an unprofessional and inaccurate way. Sep 8, 2017 at 16:19